Force and power applied to create a traveling wave

In summary: Sorry if I confused you.In summary, the author is trying to determine the power of a wavefunction on a string, but is having difficulty understanding how to do so. They find that the power is the same for both sides of the origin, and that the driving force is in the same direction for both sides.
  • #1
Miles123K
57
2
Homework Statement
A string of tension T and linear density ##\rho## is driven at ##x=0## and two travelling wave in positive and negative x-direction had been created. Calculate the force and power applied to create the travelling wave.

I am studying chapter 8 of the book the physics of waves by Howard Georgi and I am quite confused. I read the entire chapter but I still don't quite get how to acquire the travelling wave mode of a system. Can someone check my solution to this problem but also explain how to systematically work out the "travelling wave mode" from conditions provided like this question?
Relevant Equations
##\frac {\partial^2 \psi} {\partial t^2} = \frac T \rho \frac {\partial^2 \psi} {\partial x^2} ##
##F = - T \frac {\partial \psi} {\partial x}##
Again I am really confused, but I just put the traveling wave as:
##\psi(x,t) = Dcos(kx- \omega t)## for positive x
##\psi(x,t) = Dcos(kx+ \omega t)## for positive x
Then I simply differentiated and plugged in ##x=0##
##F(t) = - T D k sin(\omega t)##
and from this
## \langle P \rangle = T D^2 k \omega sin^2(\omega t)##
 

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  • #2
The string is infinite. My greatest confusion is with the process from ##A(t) = D cos(\omega t)## to the wave function. In this case I think I am just guessing the function instead of working it out. I know how to work a standing wave function from this but not a traveling wave.
 
  • #3
Miles123K said:
Then I simply differentiated
Yes, but on what basis did you choose to differentiate wrt x, not t? What were you expecting it to tell you?

At time t, what angle does the string make at the origin? What force is the tension exerting on the driver of the oscillation?
 
  • #4
haruspex said:
Yes, but on what basis did you choose to differentiate wrt x, not t? What were you expecting it to tell you?

At time t, what angle does the string make at the origin? What force is the tension exerting on the driver of the oscillation?
241313

for small ##\theta## ##sin(\theta) = tan(\theta) = \frac {\psi (a)-\psi(0)} {a}## for the limit ##a## approaches zero. The angle and force are illustrated in this image.
 
  • #5
Miles123K said:
View attachment 241313
for small ##\theta## ##sin(\theta) = tan(\theta) = \frac {\psi (a)-\psi(0)} {a}## for the limit ##a## approaches zero. The angle and force are illustrated in this image.
Good, but remember the driving force acts on both sides of the origin.
What is the speed at which the force moves the strings at the origin?
 
  • #6
haruspex said:
Good, but remember the driving force acts on both sides of the origin.
What is the speed at which the force moves the strings at the origin?
Yes. I think the wave function on the string is piece-wise and the same force drives both side. (Not very sure) The speed would just be the time derivative of ##\psi##
 
  • #7
Miles123K said:
Yes. I think the wave function on the string is piece-wise and the same force drives both side. (Not very sure) The speed would just be the time derivative of ##\psi##
Sorry, just realized that apart from the factor of 2 you have the right expression for power as a function of time; it just isn’t clear how you got it.
Not sure if the question wants the average power. That's easy enough to find.
I would take it that D and ω are inputs, so you should express k in terms of ω, ρ and T.
 
  • #8
haruspex said:
Sorry, just realized that apart from the factor of 2 you have the right expression for power as a function of time; it just isn’t clear how you got it.
Not sure if the question wants the average power. That's easy enough to find.
I would take it that D and ω are inputs, so you should express k in terms of ω, ρ and T.
Miles123K said:
⟨P⟩=TD2kωsin2(ωt)
Hey I just realized I put down the function of power not its time average. My answer for power is:
##\langle P \rangle = \frac 1 2 T D^2 k \omega = \frac 1 2 Z D^2 \omega^2##
whereas ##Z = \frac k \omega = T \frac 1 v = \sqrt {\rho T} ##

What I did is use the definition ##P = F v## and used the expression for ##F## and ##v## as stated above.

I didn't quite get what you meant by a factor of 2. Would u mind explain that?
 
  • #9
Miles123K said:
Hey I just realized I put down the function of power not its time average. My answer for power is:
##\langle P \rangle = \frac 1 2 T D^2 k \omega = \frac 1 2 Z D^2 \omega^2##
whereas ##Z = \frac k \omega = T \frac 1 v = \sqrt {\rho T} ##

What I did is use the definition ##P = F v## and used the expression for ##F## and ##v## as stated above.

I didn't quite get what you meant by a factor of 2. Would u mind explain that?
It is generating a wave each side of the origin. Same power needed for each.
 
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  • #10
haruspex said:
It is generating a wave each side of the origin. Same power needed for each.
So does that mean my solution is correct?
 
  • #11
Miles123K said:
So does that mean my solution is correct?
I believe so. It bothers me that D and ω are not given as inputs.
 
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  • #12
haruspex said:
I believe so. It bothers me that D and ω are not given as inputs.

Ummm are they not? They are provided in the question right? Or do you mean something else?
 
  • #13
Miles123K said:
Ummm are they not? They are provided in the question right? Or do you mean something else?
I based that on the synopsis you typed in post #1, but yes, I see they are given in the attachment.
 
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1. What is a traveling wave?

A traveling wave is a type of wave that moves through a medium, transferring energy from one point to another without permanently displacing the medium itself. This can be seen in phenomena such as ocean waves, sound waves, and seismic waves.

2. How is a traveling wave created?

A traveling wave is created when a force is applied to a medium, causing it to oscillate and disturb neighboring particles. This disturbance then travels through the medium, carrying energy with it.

3. What is the relationship between force and power in creating a traveling wave?

Force and power are closely related in creating a traveling wave. Force is the applied energy that causes the disturbance in the medium, while power is the rate at which this energy is transferred. In other words, the greater the force applied, the more power is required to create and sustain the traveling wave.

4. How does the direction of the force affect the traveling wave?

The direction of the force applied can determine the direction of the traveling wave. For example, if the force is applied horizontally, the resulting wave will travel horizontally. If the force is applied vertically, the wave will travel vertically.

5. Can a traveling wave be stopped or reversed?

In most cases, a traveling wave cannot be completely stopped or reversed. However, the medium through which the wave travels can affect its speed and amplitude, which in turn can alter the appearance of the wave. For example, a traveling wave can be refracted or reflected when it encounters a different medium, causing it to change direction or decrease in amplitude.

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